homomorphism

Douglas Hofstadter provides an informal definition: The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures. (Gödel, Escher, Bach, p. 49) Formally, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, ...

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Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications

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In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an antihomomorphism that is a bijection from an object to itself. In group theory, an antihomomorphism is a map between two groups that reverses the order of multiplication. So if φ : X → Y is a group antihomomorphism, φ(xy) = φ(y)φ(x) for all x,y in X. The map that sends x to x ...

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• Antihomomorphism - Examples

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In higher mathematics, "algebraic structure" is a loosely-defined phrase referring to the mathematical objects traditionally studied in the field of abstract algebra: sets with operations. The word "structure" can refer to a specific mathematical object or an even more abstract concept. For example, the monster group simultaneously is an algebraic structure, and it has an algebraic structure: the structure shared by all groups. This article uses both senses of the phrase. Algebraic structure - In the ...

Including:

• Algebraic structure - In the sense of universal algebra
• Algebraic structure - Allowing axioms other than identities
• Algebraic structure - Allowing additional structure
• Algebraic structure - Categories

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Analogy is either the cognitive process of transferring information from a particular subject (the analogue or source) to another particular subject (the target), or a linguistic expression corresponding to such a process. In a narrower sense, analogy is an inference or an argument from a particular to another particular, as opposed to deduction, induction and abduction, where at least one of the premises or the conclusion is general. The word analogy can also refer to the relation between the source and the target themselves, which is often, though not necessarily, a simil ...

Including:

• Analogy - Models and theories of analogy
• Analogy - Identity of relation
• Analogy - Shared abstraction
• Analogy - Special case of induction
• Analogy - Hidden deduction
• Analogy - Shared structure
• Analogy - High-level perception
• Analogy - Applications and types of analogy
• Analogy - Linguistics
• Analogy - Mathematics
• Analogy - Artificial intelligence
• Analogy - Anatomy
• Analogy - Law
• Analogy - Engineering

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In category theory, a zero morphism is a special kind of "trivial" morphism. Suppose C is a category, and for any two objects X and Y in C we are given a morphism 0XY : X → Y with the following property: for any two morphism f : R → S and g : U → V we obtain a commutative diagram: Then the morphisms 0XY are c ...

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• Zero morphism - Examples

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Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. Universal algebra - Basic idea. From the point of view of universal algebra, an algebra (or abstract algebra) is a set A together with a collection of operations on A. An n-ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a 0-ary operation (or nullary operation) is simply an el ...

Including:

• Universal algebra - Basic idea
• Universal algebra - Examples
• Universal algebra - Groups
• Universal algebra - Modules
• Universal algebra - Further issues

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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. In other words, it is a unital semigroup. Monoid - Definition. A monoid is a magma (M,*), i.e. a set M with binary operation * : M × M → M, obeying the following axioms: Associativity: for all a, b, c in M, (a*b)*c = a*(b*c) Identity ...

Including:

• Monoid - Definition
• Monoid - Examples
• Monoid - Properties
• Monoid - Monoid homomorphisms
• Monoid - Relation to category theory

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Alfred Tarski (January 14, 1901 in Warsaw – October 26, 1983 in Berkeley, USA) was a Polish mathematician, and widely considered one of the four greatest logicians of all time, along with Aristotle, Gottlob Frege, and Kurt Gödel. Tarski wrote on algebra, algebraic logic, measure theory, mathematical logic, set theory, and metamathematics. See Truth for a brief description of the "Convention T" (see also T-schema) standard in his "inductive definition of truth". This was an important contribution to symbol ...

Including:

• Alfred Tarski - The concept of truth in formalized languages
• Alfred Tarski - On the concept of logical consequence
• Alfred Tarski - What are logical notions?

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In mathematics, a characteristic subgroup of a group G is a subgroup H that is invariant under each automorphism of G. That is, if φ : G → G is a group automorphism (a bijective homomorphism from the group G to itself), then for every x in H we have φ(x) ∈ H: It follows that In symbols, one denotes the fact that H is ...

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Abstract algebra is the field of mathematics concerned with the study of algebraic structures called groups, rings and fields. These structures were defined formally in the nineteenth century, and, indeed, the study of abstract algebra was motivated by the need for more rigor in mathematics. The study of abstract algebra has brought into full view intricacies of the logical assumptions on which the whole of mathematics and natural science is built, and today there is scarcely a branch of mathematics which doesn't utilize the results o ...

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• Abstract algebra - History and examples
• Abstract algebra - An example

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Group isomorphism is where the objects in question are groups. Similarly, if the objects are fields, it is called a field isomorphism. In Analysis, the Legendre transform maps hard differential equations into easier algebraic equations. In universal algebra, one can provide a general definition of isomorphism that covers these and many other cases. For a more general definition, see category theory. In graph theory, an isomorphism between two graphs G and H is a bijective map f from the ...

See also:

Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications

Read more here: » Isomorphism: Encyclopedia II - Isomorphism - Applications

The following is an example of an isomorphism from ordinary algebra. Consider the logarithm function: For any fixed base b, the logarithm function logb maps from the positive real numbers onto the real numbers ; formally: This mapping is one-to-one and onto, that is, it is a bijection from the domain to the codomain of the logarithm function. In addition to being an isomorphism of sets, the logarithm function also preserves certain operations. Specifically, consider the group of positive real numbers under ordinary multiplication. The logarithm function o ...

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Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications

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Canonical is an adjective derived from canon. It essentially means "standard", "generally accepted" or "part of the back-story." basic, canonic, canonical: reduced to the simplest and most significant form possible without loss of generality, e.g. "a basic story line"; "a canonical syllable pattern" Canonical - Religion. This word is used by theologians and canon lawyers to refer to the canons of the Eastern Orthodox and Roman Catholic churches, adopted by ecumenical councils.

• Canonical - Religion
• Canonical - Literature and art
• Canonical - Mathematics
• Canonical - Computer science
• Canonical - Physics

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Isomorphism - A relation-preserving isomorphism. For example, if one object consists of a set X with an ordering ≤ and the other object consists of a set Y with an ordering then an isomorphism from X to Y is a bijective function f : X → Y such that iff u ≤ v. Such an isomorphism is called an order isomorphism. Isom ...

See also:

Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications

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The principle of compositionality has been the subject of intense debate. Indeed, there is no general agreement as to how the principle is to be interpreted, although there have been several attempts to provide formal definitions of it. Scholars are also divided as to whether the principle should be regarded as a factual claim, open to empirical testing; an analytic truth, obvious from the nature of language and meaning; or a methodological principle to guide the development of theories of syntax and semantics. The principle has been ...

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Principle of compositionality, Principle of compositionality - Critiques

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An archetypical context-free language is , the language of all non-empty even-length strings, the entire first halves of which are a's, and the entire second halves of which are b's. L is generated by the grammar , and is accepted by the pushdown automaton M = ({q0,q1,qf},{a},{a,b,z},δ,q0,{qf}) where See also:

Context-free language, Context-free language - Examples, Context-free language - Closure Properties

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Not all categories have initial or terminal objects, as will be seen below. Directly from the definition, one can show however that if an initial object exists, then it is unique up to a unique isomorphism. The same is true for terminal objects. The automorphism group of an initial (or terminal) object I is trivial. Aut(I) = Hom(I,I) = { idI }. ...

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Initial object, Initial object - Properties, Initial object - Examples

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Universal algebra - Groups. To see how this is supposed to work, let's consider the definition of a group. Normally a group is defined in terms of a single binary operation *, subject to these axioms: Associativity (as in the previous paragraph): x * (y * z)  =  (x * y) * z. Identity element: There exists an element e such that e * x  =  x  ...

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Universal algebra, Universal algebra - Basic idea, Universal algebra - Examples, Universal algebra - Groups, Universal algebra - Modules, Universal algebra - Further issues

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Because the restricted Lorentz group SO+(1, 3) is isomorphic to the Möbius group PSL(2,C), its conjugacy classes also fall into four classes: elliptic transformations hyperbolic transformations loxodromic transformations parabolic transformations (To be utterly pedantic, the identity element is in a fifth class, all by itself.) In the article on Möbius transformations, it is explained how this classification arises by considering the ...

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Lorentz group, Lorentz group - Basic properties, Lorentz group - Connected components, Lorentz group - The restricted Lorentz group, Lorentz group - Relation to the Möbius group, Lorentz group - Appearance of the night sky, Lorentz group - Conjugacy classes, Lorentz group - The Lie algebra of the Lorentz group, Lorentz group - Subgroups of the Lorentz group, Lorentz group - Covering groups, Lorentz group - Topology

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Analogy - Identity of relation. In ancient Greek the word αναλογια (analogia) originally meant proportionality, in the mathematical sense, and it was indeed sometimes translated to Latin as proportio. From there analogy was understood as identity of relation between any two ordered pairs, whether of mathematical nature or not. Kant's Critique of Judgment held to this notion. Kant argued that there can be exactly the same relation between two completely different objects. ...

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Analogy, Analogy - Models and theories of analogy, Analogy - Identity of relation, Analogy - Shared abstraction, Analogy - Special case of induction, Analogy - Hidden deduction, Analogy - Shared structure, Analogy - High-level perception, Analogy - Applications and types of analogy, Analogy - Linguistics, Analogy - Mathematics, Analogy - Artificial intelligence, Analogy - Anatomy, Analogy - Law, Analogy - Engineering

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